FQFT and cohomology
Types of quantum field thories
we have that
For that reason extended QFT is also sometimes called local or localized QFT. In fact, the notion of locality in quantum field theory is precisely this notion of locality. And, as also discussed at FQFT, this higher dimensional version of locality is naturally encoded in terms of n-functoriality of regarded as a functor on a higher category of cobordisms.
The definition of a -cobordism is recursive. A -cobordism between -cobordisms is a compact oriented -dimensional smooth manifold with corners whose the boundary is the disjoint union of the target -cobordism and the orientation reversal of the source -cobordism. (The base case of the recursion is the empty set, thought of as a -dimensional manifold.)
is an -category with smooth compact oriented -manifolds as objects and cobordisms of cobordisms up to -cobordisms, up to diffeomorphism, as morphisms.
There are various suggestions with more or less detail for a precise definition of a higher category of fully extended -dimensional cobordisms.
A very general (and very natural) one consists in taking a further step in categorification: one takes -cobordisms as -morphisms and smooth homotopy classes of diffeomorphisms beween them as -morphisms. Next one iterates this; see details at (∞,n)-category of cobordisms.
An -extended -valued TQFT of dimension is a symmetric -tensor functor that maps
smooth compact oriented -manifolds to elements of
smooth compact oriented -manifolds to -modules
cobordisms of smooth compact oriented -manifolds to -linear maps between -modules
smooth compact oriented -manifolds to -linear additive categories
cobordisms of smooth compact oriented -manifolds to functors between -linear categories
smooth compact oriented -manifolds to -linear -categories
cobordisms of smooth compact oriented -manifolds to -functors between -linear -categories
with compatibility conditions and gluing formulas that must be satisfied… For instance, since the functor is required to be monoidal, it sends monoidal units to monoidal units. Therefore, the -dimensional vacuum is mapped to the unit element of , the -dimensional vacuum to the -module , the -dimensional vacuum to the category of -modules, etc.
Here can range between and . This generalizes to an arbitrary symmetric monoidal category as codomain category.
gives ordinary TQFT.
The most common case is when (the complex numbers), giving unitary ETQFT.
The most common cases for are
, the category of -Hilbert spaces? over a topological field . As far as we know this is only defined up to .
, the category of -vector spaces over a field .
, the (conjectured?) category of -modules over a commutative ring .
See also at TCFT.
By generators and relations
By path integrals (this is Daniel Freed’s approach)
By modular tensor n-categories?
Assume is an extended TQFT. Since maps the -dimensional vacuum to as an -vector space, by functoriality is forced to map a -dimensional closed manifold to an element of . Iterating this argument, one is naturally led to conjecture that, under the correct categorical hypothesis, the behaviour of is enterely determined by its behaviour on -dimensional manifolds. See details at cobordism hypothesis.
More on extended QFTs is also at
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|
Dan Freed, Remarks on Chern-Simons theory
Daniel Freed, Quantum Groups from Path Integrals. arXiv
Daniel Freed, Higher Algebraic Structures and Quantization. arXiv
With an eye towards the full extension of Chern-Simons theory:
Dan Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups , in P. R. Kotiuga (ed.) A celebration of the mathematical legacy of Raoul Bott AMS (2010) (arXiv)